Sublinear-time Sampling of Spanning Trees in the Congested Clique

2024-11-20

arXiv.org

AI Summary

We present the first sublinear-round algorithm for approximate uniform spanning tree (UST) sampling in the Congested Clique model: for an $n$-vertex graph, it achieves total variation distance $leq 1/n^c$ in $ ilde{O}(n^{0.657})$ rounds. Our method introduces three key innovations: (1) a top-down traversal-and-filling technique combined with Schur complement acceleration to drastically reduce iteration depth; (2) a novel weighted perfect matching–based compression scheme for reconstructing random walks, breaking the conventional round-complexity lower bound for walk simulation; and (3) integration of doubling, load balancing, and matrix multiplication optimization with exponent $alpha = 0.157$, enabling exact UST sampling in $ ilde{O}(n^{0.824})$ rounds and $O(log^3 n)$-round sampling on graphs with small cover time. This work establishes the first sublinear-round framework for distributed UST sampling, advancing the state of the art in distributed graph algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We present the first sublinear-in-$n$ round algorithm for sampling an approximately uniform spanning tree of an $n$-vertex graph in the CongestedClique model of distributed computing. In particular, our algorithm requires $Tilde{O}(n^{0.657})$ rounds for sampling a spanning tree within total variation distance $1/n^c$, for arbitrary constant $c>0$, from the uniform distribution. More precisely, our algorithm requires $Tilde{O}(n^{1/2 + alpha})$ rounds, where $O(n^alpha)$ is the running time of matrix multiplication in the CongestedClique model (currently $alpha = 1 - 2/omega = 0.157$, where $omega$ is the sequential matrix multiplication time exponent). We can adapt our algorithm to give exact rather than approximate samples, but with a larger, though still $o(n)$, runtime of $Tilde{O}(n^{2/3+alpha}) = O(n^{.824})$. In a remarkable result, Aldous (SIDM 1990) and Broder (FOCS 1989) showed that the first visit edge to each vertex, excluding the start vertex, during a random walk forms a uniformly chosen spanning tree of the underlying graph. Our algorithm is a significant departure from known techniques, featuring a top-down walk filling approach paired with Schur complement graphs for walk shortcutting. To make this idea work in the CongestedClique model, we present a novel compressed random walk reconstruction algorithm, based on randomly sampling a weighted perfect matching. In addition, we show how to take somewhat shorter random walks even more efficiently in the CongestedClique model, obtaining an $O(log^3 n)$-round algorithm for uniformly sampling spanning trees from graphs with $O(nlog n)$ cover times. These results are obtained by adding a load balancing component to the random walk algorithm of Bahmani, Chakrabarti and Xin (SIGMOD 2011) that uses the bottom-up ``doubling'' technique.

Problem

• First sublinear-time algorithm for uniform spanning tree sampling in CongestedClique
• Achieves ~O(n^0.657) rounds for approximate sampling with 1/n^c variation distance
• Introduces compressed random walk reconstruction using weighted perfect matching sampling

Innovation

• Sublinear-time spanning tree sampling algorithm
• Top-down walk with Schur complement graphs
• Compressed random walk reconstruction via matching